996
J. Chem. Inf. Comput. Sci. 1996, 36, 996-1000
Topological Organic Chemistry. 10. Graph Theory and Topological Indices of Conformational Isomers Harry P. Schultz*,† Department of Chemistry, University of Miami, Coral Gables, Florida 33124
Emily B. Schultz‡ and Tor P. Schultz§ Department of Forestry and Forest Products Laboratory, Mississippi State University, Mississippi State, Mississippi 39762 Received February 27, 1996X
Distance matrices constructed from valence and vertex weighted graphs were the sources of modified information that recognized and quantified conformational isomerism exhibited by linear organic molecules. An algorithm is presented that defines the relative complexities of conformers. INTRODUCTION
The prior paper1 of this series described experiments wherein operations upon valence and vertex weighted distance matrices yielded topological indices that differentiated between geometric isomers. This present paper utilizes the same techniques used with geometrically modified topological indices to recognize and describe conformational isomerism within linear molecules. Paramount among researchers who have investigated the subtle territory of linear rotamer or conformational isomerism are Randic´, Kleiner, and DeAlba.2 Others who have directed attention to the topic of conformational analysis include Fella et al.,3 Tomczak,4 and Smellie et al.5 The superb stereochemistry compendium by Eliel et al.6 examines conformational isomerism in extensive detail. This paper records the results of experiments directed to recognizing and quantifying conformational isomerism in linear homonuclear (carbon) and heteronuclear organic molecules. Molecular topological indices described in detail in the prior papers of this seriessthe S index, product of row sums, determinant, permanent, and long hafniansare the vehicles of transmission for the conformational information. Thus, topographical ends were sought with topological means. The topic was studied in two sections. The first set of experiments was directed to assessing the influences of simple substituents on the values of conformationally modified indices of substituted ethanes. The second portion of this report examined the C4 through C7 alkane conformational isomers so penetratingly studied and described in Randic´ et al.2 COMPUTATIONS
invariant, hence descriptors, and were confirmed by use of the CIP numbering system. Valence (V) and vertex (V) weighted graphs and distance (D) matrices were utilized for all experiments, as described in detail in the preceding article1 of this series. The V weighted graphs differentiated homonuclear from heteronuclear vertices within graphs.
V ) 1 + (atomic number heteroatom atomic number carbon) V ) 1 + (atomic number heteroatom - 6) V ) C, 1; N, 2; O, 3; F, 4 V ) edge count about each hydrogen-suppressed vertex with additionally, each heteroatom unshared electron pair counted as one valence. Such V weighted graphs provided information about the bond status (single, double, triple) of each edge in a molecule as well as adding to information that distinguished homonuclear from heteronuclear vertices. The practice of projecting the conformational isomers onto a graphitic lattice described by Randic´ et al.2 was used in this work. The descriptions of rotamer status, such as synperiplanar (syn, sp) and antiperiplanar (anti, ap), were all supplanted with the succinct and unique symbols of C and T, conforming to the usages and drawings of Randic´ et al.2 The definitions of the various topological indices studied in this paper were detailed in the prior paper1 of this sequence and are briefly summarized here. The S index is the sum of the product of the valence vector and the distance matrix of order N.1 N
Molecular graphs, with hydrogen atoms suppressed and with interatomic edge counts set at unity for all atoms, were derived from the representative organic compounds listed in Tables 1 and 2. The IUPAC system of numbering and nomenclature was used in all experiments; all results were †
Present address: P.O. Box 262, Big Horn, WY 82833. Department of Forestry. § Forest Products Laboratory. X Abstract published in AdVance ACS Abstracts, July 1, 1996. ‡
S0095-2338(96)00027-3 CCC: $12.00
S ) ∑(vD)i i)1
Pettofrezzo7 defined the determinant (det) of a square matrix of order N as
det ) (-1)ka1i1a2i2...aNiN The permanent (per), sometimes referred to as the positive © 1996 American Chemical Society
TOPOLOGICAL ORGANIC CHEMISTRY
J. Chem. Inf. Comput. Sci., Vol. 36, No. 5, 1996 997
Table 1. Descriptors of Selected Substituted Ethanes Derived from the Vertex and Valence Weighted Distance Matrices and Modified to Represent the Individual syn (C) and anti (T) Conformations of Each Graph index DNN(GVVw) no.
compound
1
butane C T 1,3-butadiene C T 1,2-diaminoethane C T 1,2-diaminiumethane ion C T 1,2-ethanedialdiimine C T 1,2-ethanediol C T 1,2-ethanedial C T oxalic acid C T 1,2-difluoroethane C T
2 3 4 5 6 7 8 9
a
a
RMN
+26 -26 +42 -42 +44 -44 +32 -32 +66 -66 +74 -74 +102 -102 +190 -190 +116 -116
S
PRS
detb
per
1haf
44 70 18 120 162 78 128 172 84 56 88 24 288 354 222 356 430 282 648 750 546 1874 2064 1684 800 916 684
2 304 2 330 2 278 20 736 20 778 20 694 36 864 36 908 36 820 9 216 9 248 9 184 186 624 186 690 186 558 186 624 186 698 186 550 746 496 746 598 746 394 133 885 737 216 133 885 737 406 133 885 737 026 589 824 589 940 589 708
-48 74 22 -432 474 390 -768 812 724 -192 224 160 -3 888 3 954 3 822 -3 888 3 962 3 812 -15 552 15 654 15 450 -14 929 920 14 930 110 14 929 730 -12 288 12 404 12 172
256 282 230 2 304 2 346 2 262 4 096 4 140 4 052 1 024 1 056 992 20 736 20 802 20 670 20 736 20 810 20 662 82 944 83 046 82 842 2 058 089 472 2 058 089 662 2 058 089 282 65 536 65 652 65 420
104 130 78 936 978 894 1 664 1 708 1 620 416 448 384 8 424 8 490 8 358 8 424 8 498 8 350 33 696 33 798 33 594 159 376 896 159 377 086 159 376 706 26 624 26 740 26 508
Rotamer modification value. b Absolute values used in calculation of modified indices.
Table 2. Descriptors Derived from the Vertex and Valence Weighted Distance Matrices of Linear C4-C7 Alkanes Modified To Represent the Various Syn (C) and Anti (T) Conformations of Each Graph index DNN(GVVw) no.
compound
1
butane C T pentane CC CT TT hexane CCC CTC TCC TTC TCT TTT heptane CCCC CTCC TCCC CTTC TTCC TCTC TCCT TTTC TTCT TTTT
2
3
4
a
RMVa +26 -26 +88 0 -88 +202 +78 +62 -62 -78 -202 +384 +208 +176 +32 0 0 -32 -176 -208 -384
S
PRS
detb
per
1haf
44 70 18 100 188 100 12 190 392 268 252 128 112 -12 322 706 530 498 354 322 322 290 146 114 -62
2 304 2 330 2 278 235 200 235 288 235 200 235 112 35 283 600 35 283 802 35 283 678 35 283 662 35 283 538 35 283 522 35 283 398 7 326 498 816 7 326 499 200 7 326 499 024 7 326 498 992 7 326 498 848 7 326 498 816 7 326 498 816 7 326 498 784 7 326 498 640 7 326 498 608 7 326 498 432
-48 74 22 256 344 256 168 -1280 1482 1358 1342 1118 1102 1078 6144 6528 6352 6320 6176 6144 6144 6112 5968 5936 5760
256 282 230 9 216 9 304 9 216 9 128 549 952 550 154 550 030 550 014 549 890 549 874 549 750 42 720 256 42 720 640 42 720 464 42 720 432 42 720 288 42 720 256 42 720 256 42 720 224 42 720 080 42 719 048 42 719 872
104 130 78 1 448 1 536 1 448 1 360 49 760 49 962 49 838 49 822 49 698 49 682 49 558 1 000 000 1 000 384 1 000 208 1 000 176 1 000 032 1 000 000 1 000 000 999 968 999 824 999 792 999 616
rel potential energy, kcal/mol 6.1 0 12.0 6.0 0 12.0 11.9 6.0 5.9 0 17.9 12.0 11.9 11.9 11.8 6.0 5.9 0
Rotamer modification value. b Absolute values used for calculation of modified indices.
determinant, was described in detail by Minc.8
per ) a1i1a2i2...aNiN The product of the row sums (PRS) was an early, rough approximation to the permanent. Caianiello9 described the hafnian (1haf) as the summation over all permutations i1, i2,
..., i2N which satisfies the limitations i1 < i2, i3 < i4...i2N-1 < i2N; i1 < i3 < i5... < i2N-1. These five indices were the carriers of the conformational information presented in our paper. The geometries about a pair of single bonded vertices responsible for the conformational isomerism were indicated by rotamer factors (RF) of +1, +1 for C geometry; -1, -1 for T geometry. All other vertices of the graphs were
998 J. Chem. Inf. Comput. Sci., Vol. 36, No. 5, 1996
SCHULTZ
ET AL.
Table 3. Conformational Sequence Comparisons of Linear C4-C7 Alkanes as Projected onto a Graphite Lattice Randic´ et al.,2 increasing sequences molecular ID numbers C4
C5
C6
C T
CC CT TT
CCC TCC CTC TCT TTC TTT
φ Values C7
CTCC TCCC TCCT TTCC CTTC TCTC TTCT TTTC TTTT
C4
C5
C6
C T
CC CT TT
CCC TCC CTC TTT TCT TTC
assigned RF values of zero. These factors were used to calculate a rotamer modification number (RMN) which was then incorporated into a topological index, using the same principles as described in the previous paper.1 A specific example is pictured using 1,2-difluoroethane (Table 1, 9). F
C
C
F
1
3
4
2
CIP numbers
3
1
2
4
IUPAC numbers
4
1
1
4
vertex (V) weights
4
2
2
4
valence (v) numbers
0
±1
±1
0
rotamer factors (RF)
0
±58
±58
0
rotamer modification numbers (RMN) vide infra
The VV weighted D matrix was constructed
[
] [ ]
0 2 D44(GVVw) ) 16 32 which was then transposed
D44(GVVw)T
0 2 ) 2 4
2 0 32 16
2 4 0 48
4 2 48 0
2 0 4 2
16 32 0 48
32 16 48 0
and the two matrices were summed, as explained, and justified in the preceding publication.1 In heteronuclear graphs the greater weights of the heteronuclear vertices placed the bulk of the information in rows outside of the conformational rows, as seen in rows no. 3 and no. 4 of the VV weighted D matrix of 1,2-difluoroethane. The sum of the D and DT matrices provided relatively more information in rows no. 1 and no. 2, the conformational centers of the compound. The technique also provided a more equitable comparison between homonuclear and heteronuclear graphs.
[
0 4 T 4 4 D4(GVVw) + D4(GVVw) ) 18 36
4 0 36 18
18 36 0 96
36 18 96 0
]
The RF values for the C and T conformations of 1,2-
this paper, decreasing sequences all index values C7 TCCC CTCC TCCT TTCC CTTC TCTC TTCT TTTC TTTT
C4
C5
C6
C7
C T
CC CT TT
CCC CTC TCC TTC TCT TTT
CCCC CTCC TCCC CTTC TTCC TCTC TCCT TTTC TTCT TTTT
[ ] [ ]
difluoroethane were built into two diagonal matrices.
+1 0 0 0 0 +1 0 0 D44(RF, C) ) 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 D44(RF, T) ) 0 0 0 0 0 0 0 0 Premultiplication of the D + DT matrices with the diagonal RF matrices gave the information for the rotamer modified numbers (RMN) of the two conformers of 1,2-difluoroethane.
[ ]
0 4 18 36 4 0 36 18 C) ) 0 0 0 0 0 0 0 0 0 -4 -18 -36 -4 0 -36 -18 D44(RMN, T) ) 0 0 0 0 0 0 0 0 The content of each matrix was summed. The rotamer modification number (RMN) for the S index of the C rotamer of 1,2-difluoroethane was +116; that for the T conformer was -116. These modifications were added to the S index value (800) of 1,2-difluoroethane to give the C conformer modified value of 916 and the T modified value of 684 for the S indices of the two rotamers of 1,2-difluoroethane. The other topological indices were calculated in the same fashion. All five molecular topological indices reported in this paper exhibited the same arrangements, but the larger index values possessed a lower ratio of response to the rotamer modification values. All the substituted ethane derivatives of Table 1 have only one single, central bond about which rotamer modification exists. In the instance of larger molecules, such as pentane (Table 2, 2), the various paired centers of conformational isomerism were algebraically summed to arrive at RF values for each vertex. For example, the CC conformation of pentane displays a set of rotamer factor values (RFV) calculated as follows. D44(RMN,
2,3 C 3,4 C
+1
RFV
+1
[
]
+1 +1
+1
+2
+1
The all-C conformations of all normal C4-C7 alkanes were included in this study, even though in the larger molecules
TOPOLOGICAL ORGANIC CHEMISTRY
J. Chem. Inf. Comput. Sci., Vol. 36, No. 5, 1996 999
Table 4. Compaction Indices (CI)a for All Conformers of Normal C4-C7 Alkanes Superimposed onto Graphitic (G) and Pentagonal (P) Lattices conformer C4 C5 C6
C7
C T CC CT TT CCC TCC TTC CTC TCT TTT CCCC TCCC TTCC CTTC TCCT CTCC TTCT TTTC TCTC TTTT
G
P
0.88 0.25 2.01 0.58 0.29 5.22 1.59 0.94 0.67 0.42 0.28
1.18 0.31 3.44 0.73 0.38
3.42 2.03 1.72 1.68 1.15 0.87 0.71 0.54 0.29
2.12 1.17 0.86 0.53 0.35 2.34 2.27 2.42 1.55 1.09 0.85 0.65 0.38
a
CI ) sum vertical distances of vertices from baseline/baseline length C1-CN.
whose path graphs were imbedded in the honeycomb lattice the terminal vertices would curl around and superimpose on the first carbon atoms of C6 and larger compounds. Although structurally meaningless, the mathematics involved is rational, and in such instances one could in a practical vein imagine a very tight spiral. The conformers were also pictured by imprinting each rotamer on a regular pentagon lattice; the interior angles of 108° of the pentagon are very nearly of the same value as the normal tetrahedral valence angle of 109.5°. Randic´ et al.2 have pictured on pages 278 and 280 of their presentation, conformations of the C4-C7 normal alkanes superimposed on graphitic lattices. RESULTS
Table 1 lists a series of related ethane derivatives, with the no. 1 and no. 2 vertices successively substituted with C, N, O, and F atoms, at increasingly higher carbon oxidation levels for N and O. Note the different index values exhibited for the two congeneric graphs of butane and 1,2-diaminiumethane ion (Table 1; 1, 4) as well as the degenerate PRS and long hafnian values for glyoxal diimine and ethylene glycol (Table 1; 5, 6). No regard was given to the additional geometric isomerism extant in the trigonal hybridized imino groups of 1,2-ethanedialdiimine (Table 1, 5). Because, as Randic´ et al.2 observed, most thermodynamic physicochemical properties belong to a mixture of conformations, not to a single rotamer, the order of progression of the index values for the various alkane rotamers listed in Table 2 was contrasted with the data of Randic´ et al.,2 the latter being the authoritative basis for the comparisons. Table 3 shows the sequence relations, from all-C to all-T rotamers, for the two Randic´ sets of values,2 molecular ID numbers, and the folding (φ) values, compared with the conformational sequences arrived at in this study. As expected, all sequences were the same for butane and pentane but differed for hexane and heptane in all sets of comparisons. These latter series were similar in that both sequences, with one exceptionsthe TTT location in the Randic´ φ series for hexanessubdivide each progression into
Figure 1. Possible conformations of seven-vertex chains superimposed on a pentagonal lattice. See also Figure 2, ref 2.
about equal thirdssmostly C bonds, equal numbers of C and T bonds, and mostly T bonds. The Randic´ numbers, however, unlike one of this paper’s sequences (TTCC ) TCTC, for heptane), have no degenerate values. It is interesting to observe that the identical topological index values reported in this paper for the heptane rotamers TTCC and TCTC have one each of interior and exterior T and C sets of values but that the compaction indices (CI, vide infra) are different, indicating dissimilarity between the two conformers. In order to ascertain the degree of similarity/dissimilarity among the rotamers, a compaction index (CI) was devised. The CI index was graphically (10 cm ) one unit length) determined by dividing the distance (the baseline) between the two termini of a conformer into the sum of the vertical distances from the baseline, extended if necessary, of all vertices of a graph. The lower the CI quotient, the less puckered was the conformer, with all-T conformers having the lowest CI values and all-C rotamers the highest CI numbers. Graphs with similar quotients were conformationally more similar to one another than graphs that possessed widely divergent CI quotients. Table 4 summarizes these data, and also includes data obtained from graphs superimposed onto pentagonal lattices. As expected, the pentagonal lattice CCC rotamer of hexane and the TCCC and CCCC rotamers of heptane fold the initial and terminal vertices together, and hence their compaction indices are meaningless. In the instance of the heptane, TCCT, CTTC, and TTCC conformers, a different sequence of CI numbers was observed between values derived from the graphitic and pentagonal lattices. Although the same general similarity/ dissimilarity comparisons were observed as were pictured in the Randic´ Figures 5 and 6, there was not an exact parallelism with the data printed in the Randic´ Table 7. Additionally, there was only modest agreement between sequences pictured in Tables 3 and 4 of this paper. As a final experiment the rotamer modified topological indices were compared with the rotational energy barriers of the various conformers studied. From a plethora of data, Eliel et al.6 designate the syn/anti energy difference about the C2-C3 bond of butane as lying between 4.5-6.1 kcal/ mol. The data, gathered from a variety of spectrographic, molecular mechanics, and ab initio sources, were focused mainly on the methyl-methyl interactions about the middle
1000 J. Chem. Inf. Comput. Sci., Vol. 36, No. 5, 1996
bond of butane. To more completely utilize such data the work of Wiberg and Murcko10 was additionally searched and averaged to provide from one source all the energy data (kcal/ mol) used for these syn/anti group interaction calculations. The following values (kcal/mol) were used in this study: methyl-methyl, 6.1; methylene-methylene, 5.9 (about the hexane C3-C4 eclipsed bond); methylene-methyl, 6.0 (by interpolation of the above values). The calculated syn/ anti relative potential energy data for the various conformers of butane, pentane, hexane, and heptane are listed in Table 2. Figure 1 pictures possible conformations of heptane; stepwise removals of its terminal vertices progressively form, in turn, the rotamers of hexane, pentane, and butane. The terminal vertex overlapping of the CCC rotamer of hexane and the CCCC and TCCC conformers of heptane afford meaningless information, hence their energy data are omitted from Table 2. Table 2 shows the sequence parallelism between rotamer modified indices and relative potential energy contents of all conformers of butane, pentane, hexane, and heptane, including the identical energy values for the TTCC and TCTC rotamers of heptane, the two rotamers that presented degenerate modified topological indices. A quantitative assessment of the relationship between the modified index value and the energy content of each conformer was attempted. The S index was examined, for it was the easiest value to arrive at; the determinant index was analyzed, because in past studies of this series it generally gave the best statistical results. First, all the collective data from all points reflected by butane through heptane rotamers were studied; no relationship was observed in the instances of using modified S or determinant index data. Second, the hexane and heptane sets of rotamers were separately examined. (Too few data points were available in the smaller butane and pentane portions of data.)
S (D) Index (Table 2, column 4) hexane E, kcal/mol ) 0.6404 + 0.04358(S) n ) 5 r2 ) 0.996 s ) 0.3579 F ) 778.49 heptane E, kcal/mol ) 2.0816 + 0.03013(S) n ) 8 r2 ) 0.990 s ) 0.5765 F ) 622.32
SCHULTZ
ET AL.
The heptane S and determinant modified index comparisons gave similar equations and the same statistical data. These results are tempered by the caveat that some of the conformers would have had 1,5 terminal carbon atom repulsions, as seen in the rotamers of CC pentane, TCC hexane, and TTCC and TCCT heptane, although calculations by Eliel et al.6 indicate less than 0.4% of the high energy g+g- rotamer of pentane is present in its conformational equilibrium. When, however, in spite of this circumstance, the Eliel et al.6 range of 3.3-3.7 kcal/mol was included to reflect the energy of the 1,5 dimethyl repulsions, the hexane and heptane conformational energy sequences were thrown completely awry. This was to be expected, for each modified topological index represented only the conformational status along the bonds of the alkane chain and not across its nonbonded ends. CONCLUSIONS
A technique has been presented that identifies and quantifies the molecular topological indices of the various conformational isomers of the C4-C7 normal alkanes. The ease of execution of the procedures described in this paper presents an interesting mode of applying topological techniques to attain topographical ends. ACKNOWLEDGMENT
The authors thank the referees for constructive and positive comments about this paper. REFERENCES AND NOTES (1) Schultz, H. P.; Schultz, E. B.; Schultz, T. P. Topological Organic Chemistry. 9. Graph Theory and Molecular Topological Indices of Stereoisomeric Compounds. J. Chem. Inf. Comput. Sci. 1995, 35, 864-870. (2) Randic´, M.; Kleiner, A. F.; DeAlbe, L. M. Distance/Distance Matrices. J. Chem. Inf. Comput. Sci. 1994, 34, 277-286. (3) Fella, A. L.; Nourse, J. G.; Smith, D. H. Conformation Specification of Chemical Structures in Computer Programs. J. Chem. Inf. Comput. Sci. 1983, 23, 43-47. (4) Tomczak, J. An Explicit Representation of Molecular Geometry and Topology for Small Molecules. J. Chem. Inf. Comput. Sci. 1995, 35, 199-202. (5) Smellie, A.; Kahn, S. D.; Teig, S. L. Analysis of Conformational Coverage. 1. Validation and Estimation of Coverage. J. Chem. Inf. Comput. Sci. 1995, 35, 285-294.
det (D) Index (Table 2, column 4)
(6) Eliel, E. L.; Wilen, S. H.; Mander, L. N. Stereochemistry of Organic Compounds; Wiley-Interscience: New York, 1994; pp 597-834.
hexane
(7) Pettofrezzo, A. J. Matrices and Transformations; Prentice-Hall: Englewood Cliffs, NJ, 1966; pp 13-29.
no fit
(8) Minc, H. Permanents. Encyclopedia of Mathematics and Its Applications; Rota, G.-C.; Ed.; Addison-Wesley: Reading, MA, 1978; Vol. 6, pp 1-119. (9) Caianiello, E. R.; Proprieta` Pfaffiani e Hafniani. (Properties of Pfaffians and Hafnians). Recerca, Napoli 1956, 7, 25-31.
heptane E, kcal/mol ) -173.35 + 0.03013(det) n ) 8 r2 ) 0.990 s ) 0.5765 F ) 622.32
(10) Wiberg, B. K.; Murcko, A. M. Rotational Barriers. 2. Energies of Alkane Rotamers. An Examination of Gauch Interactions. J. Am. Chem. Soc. 1988, 110, 8029-8038.
CI960027V
